TPA TRACON Controller reported an airborne conflict between an aircraft climbing out of SPG and an aircraft on approach to Runway 36 at PIE airport.

Narrative

Aircraft X was cleared for a VFR GPS Runway 7 approach to a full stop at SPG. Aircraft Y was on a base at 2;000 ft to Runway 36 at PIE. SPG Tower authorized Aircraft X to do the published miss without coordination. They turned directly into Aircraft Y at 2;000 feet. I had to break Aircraft Y off approach and issue a traffic alert. Both aircraft got about 1.25 miles and same altitude opposite direction. SPG is not authorized to issue the published miss without coordination. SPG Tower said that we have no control there because they are outside Class B. This also brought the aircraft into TPA Bravo [airspace] for a short period of time. Many aircraft depart out of SPG west bound through the PIE final. This is very dangerous. Especially with increased traffic at PIE. The Class B needs to be moved further west to ensure planes landing PIE don't crash into VFR aircraft off SPG. It is only a matter of time before a midair collision happens here. Something needs to be done. People are going to die. This is an issue that has been reported many times. Please help us fix this. Please stop letting it go.